# Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding EMM $Q^B$. From Björk's book I have that $$\frac{S(t)}{B(t)}+\int_0^t \frac{dD(s)}{B(s)}$$ must be a martingale under $Q^B$ (actually this is not the entirely true for all cases, but if $B$ only has the dt-term it will be). Then I could find the dynamics of $S$ under the new measure by first finding the dynamics of the expression above, i.e. $$d\Big(\frac{S}{B}\Big) + \frac{dD}{B}$$ using Ito's formula. When the dynamics are in GBM-form I will neatly be able to factor out the $S/B$-term. Then, after using Girsanov Theorem, $dW = \varphi + dW^{Q^B}$, I would end up with a dt-term $$\frac{S}{B}\big((\alpha - r + \delta) + \varphi (\sigma +\delta)\big)$$ which must be equal to 0 since it should be a martingale. Here I have assumed the following dynamics of the processes

$dB = rBdt$

$dS= \alpha S dt + \sigma S dW$

$dD = \delta S dt$

Then I could easily solve for $\varphi$ and then the dynamics under the new measure follows. However, now assume the stock is traded on a foreign market with corresponding exchange rate $E$, then I'd like to determine $S_d$ (which is the process of $S$ in domestic currency) defined by $S_d = S \cdot E$. Analogously we have the dividend process $D_d$. In this case $S_d$ will still have dynamics on GBM-form with different parameters. The $D_d$ term will change quite dramatically though (I get $dD_d = (\delta S_d + D_d \mu_E)dt + \sigma_E D_d dW$, where $\mu_E, \sigma_E$ are the parameters in the GBM of the exchange rate). Since not all terms include $S_d$ I will not be able to factor that out when I solve for $\varphi$ as in the previous case. So I can't find the new dynamics in this manner. Is there something I'm missing or could I approach this problem differently?

Edit: Maybe it would be easier to find the dynamics of just $S$ and $E$ under the new measure, and then find the dynamics of $S_d$ by $S_d=S \cdot E$? I could find the dynamics of $E$ by the fact that $E/B$ must be a martingale under the measure (is this true?).

• Ok, I just realized that the dividends process in domestic currency that I use is not correct. $D \cdot E$ is just the accumulated dividends process in foreign currency, evaluated in domestic currency. This is not really what I wanted. The correct dynamics are $dD_d = \delta S_d dt = \delta E \cdot S dt$, i.e. the dividends process received in domestic currency at each infinitesimal time point. Then the equations will solve nicely. I guess it is not possible to find an EMM under those conditions I had firstly. – Good Guy Mike Feb 21 '14 at 15:58
• could you perhaps edit the question accordingly ;) – Probilitator Feb 22 '14 at 11:28